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Tuesday, September 3, 2013

Building Functions, Clarified

So, I really appreciate all the thoughtful input on this previous post. I started commenting, but the comment got real big.

To give some background, I'm writing a unit flow for an introduction to quadratics unit. The big things I'd like students to remember from this unit for a long time are:

some situations that can be modeled with quadratics, i.e. falling objects, triangular numbers, area

compare/contrast with linear and exponential (which come before this). how can we tell if given information (situation, table, graph, 3 points) can be modeled with linear, exponential, or quadratic?

what information can easily be obtained from each of the forms (vertex, standard, factored), given an equation

what form (vertex, standard, factored) is it easiest to write an equation in, given different information

why the graphs look like that (why symmetrical? why a U-shape? why does it have a max or min?)

What I DON'T want to do in this unit is algebraically convert between equivalent forms, or solve quadratics with various techniques.

The big question is still, how do you introduce things that are more complicated than ax2?

10 comments:

Option #1 does not directly reveal any symmetry. Wouldn’t it be better to include points on the other side of the vertex, or the graph itself?

Option #2: Would people consider introducing linear equations this way? y = mx + b with some sliders – see how m & b make it different. If not, why do it for quadratics? I like sliders and questions like option #2, but not as an introduction.

I agree with Michael. Haven't seen CME's version of function transformations, but I have an idea for how to teach it: encryption.

Say you create a code in which 1 = A, 2 = B, etc. That code is f(x). But you want your code to change every day, so it's harder to break. So you introduce a constant in your code, making it f(x - h), where h changes every day according to some plan you've communicated in advance to your accomplices.

Have students encrypt messages such as "MTBoS" using different values of h. For example, f(13) = M, so to translate the message "M", you send the number 13. But if h = 2, the you have to use f(15 - 2) to get M, so you send the number 15.

You can even make a graphical connection by representing the original code on a grid, with 1, 2, 3, ... along the bottom axis and A, B, C, along the vertical axis. Just shade in the little squares for (1, A), (2, B), etc. Or you could make up a different original code such as (1, Z), (2, Y),...

Regardless, students can see what happens to the graph when the daily code switches from f(x) to f(x - 2): the whole graph shifts by 2. But they can discover that fact by trying to encrypt a message and finding out that each input number they use needs to be 2 units greater than it used to be.

This would also allow you to bring in the concept of a function vs. relation, because messages encrypted with non-functions wouldn't be able to be decrypted well.

I've been meaning to work on this approach for a while, but right now I have other projects going on, so thought I'd mention it to everyone else here.

I'm kind of intrigued by an intro to quadratics that asks students to graph sums and differences of linear functions represented as lines (ideally on unitless axes so no equations). And then to have them graph the product of two linear functions represented as lines. You don't get all quadratics that way but it's a neat introduction to the family. From there transformations have a kind of meaningful connection. Plus, the Zero Product Property jumps right out at you.

Hi Kate,What about having them find the rule in a pattern such as the triangle numbers (presented visually)? That gives a quadratic function that is a bit more complex, and it could be used to show the idea of a rate of change that is increasing, as you are adding an increasing amount each time to get the next number.

If what Dan said sounds appealing, there are lots of good pattern problems that take a quadratic form when the data points are graphed. My favorite is one Ellen Kaplan did, that I began to call the Magic Pancake problem when I started doing it. My write-up is in the bottom half of this post.

So, (Dan/Sue) what you guys suggesting is how we would get to option 1.

But I think I don't like option 1. Reason being, it naturally lends itself to a desire/need to show whether or not expressions are equivalent. One kid comes up with (x-3)^2, another kid comes up with x^2-6x+9, so who's right? And, manipulating algebraic expressions is not something I want to get into in this unit.